Question

the function $f(x)$ is graphed below. write the domain and range using inequality notation. $f(x) = -|x + 4| + 5$ draw show your work here domain: range:

Explanation:

Step1: Determine the domain of the absolute value function

The function \( f(x) = -|x + 4| + 5 \) is an absolute value function. The domain of any absolute value function (or any linear function inside the absolute value) is all real numbers because there are no restrictions on the values of \( x \) that we can plug in. In inequality notation, this means \( -\infty < x < \infty \) or more commonly written as \( x \in (-\infty, \infty) \) but in inequality notation for all real numbers, we can express it as \( -\infty < x < \infty \) or using the standard notation for all real numbers \( x \) such that \( x \) is any real number, which in inequality form is \( -\infty < x < \infty \). But usually, for absolute value functions, the domain is all real numbers, so in inequality notation, we write \( -\infty < x < \infty \) or \( x \) can be any real number, so \( -\infty < x < \infty \).

Step2: Determine the range of the absolute value function

The general form of an absolute value function is \( y = a|x - h| + k \). In our function \( f(x) = -|x + 4| + 5 \), \( a=- 1 \), \( h = - 4 \) and \( k = 5 \). The coefficient \( a=-1 \) is negative, so the graph of the absolute value function opens downwards. The vertex of the absolute value function \( y=a|x - h|+k \) is at \( (h,k) \), so the vertex here is at \( (-4,5) \). Since the graph opens downwards, the maximum value of the function is at the vertex, which is \( y = 5 \), and the function will take all values less than or equal to 5 as \( x \) varies over all real numbers. To find the range, we note that the absolute value \( |x + 4| \geq 0 \) for all real \( x \). Multiplying both sides by - 1 (and reversing the inequality sign) gives \( -|x + 4| \leq 0 \). Then adding 5 to both sides gives \( -|x + 4|+5\leq5 \), so \( f(x)\leq5 \). Also, as \( x \) approaches \( \pm\infty \), \( |x + 4| \) approaches \( \infty \), so \( -|x + 4| \) approaches \( -\infty \), and thus \( f(x)=-|x + 4| + 5 \) approaches \( -\infty \). So the range of the function is all real numbers \( y \) such that \( y\leq5 \), which in inequality notation is \( -\infty < y \leq 5 \).

Answer:

Domain: \( -\infty < x < \infty \)

Range: \( -\infty < y \leq 5 \)